Offline Traffic Engineering Design and Optimization in Communication Networks with Risk Analysis

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日本オペレーションズ。リサーチ学会 2005年春季研究発表会

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0軋meⅦra侃cⅡmg五mee『五mgⅡ）es五gmamdOp七五m五za七五①mim

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中科院数学院

＊呉軍 WUJun

OllO7734 甲南大学

岳五一 YUEWuyi

中科院数学院

江寿陽 WANGShouyang

where∧representstochoose thesmalleronebe−

tweentwocomponents，and＋representstochoose thepositivepart・

Let rI（b，D）denote the mean profitfunction

withthelincarpcnaltycostasfo1lows：

n（岬＝γ上b榊）血川上＋∞′（埴

1Int．roduction

In［1】a．nd［2】，thestochastic trafBcengineer− ingproblemshavcbcenstudicd・In［3］，WCprC−

Sented an analysisfor theloss rate constraintin

SuChproblemsandshowedtheimpactoftheloss

rateconstraintonthenetworkperformancebynu−

mericalresults．Inthi＄paper，WepreSentananaト

ysisforthcmodclprcscntcdin［3］toderivethc

Optimalbandwidthcapacitywithalinearpenalty

COSt．We alsoanalyze theriskofnetwork pro飢 Shortfa11byuslngmean−Varianceapproach・

2 SystemModel

A Communication Network（CN）should de−

riveitsrevenuebyserv1ngtra氏cdemandincluding

VOice，PaCkct data，imagc andfu11−mOtion vidco・

FbrunitbandwidthcapacityallocatedtotheCN，

anunitcostwillbecharged・Fbrunsatisfied11nit

t．ra庁icdemandwiththelimitationofnetworkband−

Width，alinearpenalty cost wi11beaddedinthe Objectivefunction．Theobjectiveofthissystemis

tomaximizetheCNmeanprofit．

Uncertain totaltra侃c demandin t，he CN de−

noted by D ＞ Ois characterized by a random

distributionwiththeprobabilitydensityfunction

f（x）andthecumulativedistributionfunctionF（x）．

Let b＞Odenote theamount ofbandwidthca−

pacity provisionedin the CN，r denote the unit

revenueoftheCNbyservlngthetrafRcdemand，

Cdenote theunit costfor unitbandwidthcapac−

ity allocatedin the CN，and q denote thelinear

penaltycostfortheunsatis丘edunitdemand．Let P（b≧6D）≧1−∈denotethelossrateconstraint，

andlet Ch。X ＞O denote the maximalcapacity

t．hat can bca1locatcdin thc CN．Tb avoid unrcal− isticcases，WemakethefollowlngaSSumptions：（1）Systemparametersare：r＞q＞0，r＞c＞0．（2）Lossrateconstraintpa，rameteris：0≦6≦1．（3）Confidencelevelis：0≦1−e≦1．

3 0p七imal1Bandwid七hAlloca七五onw五th

Pema丑tyCost Let7r（b，D）denotetherandomprofitfunction byservlngtrafRcdemandinthenetworkwiththe linea．rpenaltycost，namely，打棒，刀）＝γ・（わ∧β）−q岬一切＋−Cゐ（1）＋00 上（ェーわ）J（霊）血−Cむ・（2）￣q Theobjectivefunctionofthesystemis 口＊＝ロ（ぁげ）），（3） SubjecttoP（b≧6D）≧1−∈andb≦Cha，：・Ⅰ㌣is theoptimalpro丘tfunction・

Next，We analy21e the property of tlle mean

profitfunctionⅢ（b，D）．Thefirstorderderivative OfⅢ（b，D）withrespecttobisgivenasfollows： dⅢ（む，β）＝（r＋q−C）−（γ＋q）ダ（町（4） dむ ThesecondorderderivativeofrI（b，D）withre− SPCCttObisglVCnaSfo1lows： d2Il（♭，か）＝−（γ＋ヴ）J（わ）≦0．（5） dわ2 Therefore，WeCanSaythatⅢ（b，D）isaconcave functionofb．So，thcoptimalbandwidthcapacity _︵︵一1 ダ ∫ Or

守），Whereダ￣1（・）

without constraints is is theinversefunction Wedefinethelossrateastheprobabilitythat

thetra氏cdemandcannOt beserved by the CN． Thercforc，thclossratcconstraintisequlValcntto

b∈【6F−1（1−e），＋∞）．Ifweconsiderthemax−

imalcapacity constraint，the optimalbandwidth

CapaCityfortheCNcanbeglVenby

C

γ＋q−

∨紺￣1（1−∈） _{∧q乃α。（6）} γ＋q

Where v represents to choose thelarger orle be− tweentwocomponents．

Weconsiderafu11ydistributednetwork，Where

the tra且c demandis a5Sumed tofo1low auniform

distributionas aspccialcxampIc．Wcgivcsomc

nllmericalresults to show theimpact ofpenalty

−272−

(2)

ヰ去（q＋γ）2れ（2γ2＋2q2＋血汀−3γC −39C＋c2）か（去q2＋言qr）わ2＋吉和（10）

WiththesamesystemparametersforFig・1，

We glVe SOme numericalresults to show theim− PaCtOfriskaNerSeneSS

．

thcincrca5C Of risk avcrscncss．Ordinate axis of

Fig・2correspondstothepercentagedi鮎renceof

theobjectivefunctionftomtheobjectivefunction

without risk． OStOnthcbandwidthcapacity（sccFig・1）・Hor−

1ZOntalaxisq／r ofFig．1corresponds to thein− CreaSe Ofpenalty cost．Ordinate axis ofFig．1

COrreSpOndstothepercentagedi鮎renceoftheop−

timalbandwidthcapacityfromtheoptimalband−

width capacity without penalty. For comparing

withthemodelof［1］，示echoosesystemparame− tersfortwocasesasfo1lows：（1）r＝7．5，C＝1．5， and（2）r＝7・5，C＝0・5，intheinterval［0，1］・亜凡∵仇乱〓机正二n仇盲8￡彗とd∞ ＝：；…￡… 皿孤皿岬仙狐几 u点︸リ5﹂電せ艮一〇

十l亡75c＝1．5 ↓戸尋．5c＝α5

nO 81 02 0．3 0．4 0．5 8￠ 0．7＋0月 0，9 1．0 如I∼Co雨 Figurel：Impactonbandwidthcapacity． ThenumericalresultsshowninFig．1，includ一 ngtheresultspresentedin［1］，reVealadistinct

lmpaCtOfpenaltycost on the CN bandwidthca−

pacity．Withthcsamcpenaltycost，thclargerthe unitcostcis，thegreaterth

Widthcapacityis．

4 RiskAnalysiswithPenalty Cost DuetotheuncertaintrafRcdemand，theprofit

isalsouncertain，Wedefinetheriskasthedeviation

fromtheoptimalprofitinthispaper．

Weanalyzctheriskofprofitshorthllbyuslng themean−Varianceapproach．Theobjectivefunc− tion，Whichisdenotedby◎＊，isglVenaSfo1lows： ◎＊＝叩げ）−αⅤ叫咄β）］）（7） Whereα（0≦α≦1）isariskaverseness ．（1），andVar［7T（b，D）］isthcvarianccof7T（b，D）． Whenthetrafncdemanddist．ributedint，hewhole networkis assumedtofo1lowtheuniformdistribu＿

tion，WeCanObtainthemeanfunctionby

Ⅲ（岬）＝一宇紬伸一C）わー芸・（8）

ThevarianCefunctionisobtainedby

叫棉卯＝−（q＋γ）2れ（2γ2＋2q2＋c2

埼r−3γC−3qc）か（ヴ2＋蝉2＋れ9）

Moreover，theobjectivefunction◎■presented inEq・（7）isgivenby 竺

◎＊＝（ト距（叫−C）む−］

0．0 0．10．2 0．3 0．4 0．5 0．8（‖ 0．も 0．9l．O Ris一触lちeneSSP8ralllOter Figurc2：Impactonobjcctivcfunction． ThenumericalresultsshowninFig・2，includ一 ngtheresultspresentedin［1］，reVealadistinct

lmpaCtOfriskonthcCNobjcctivcfunction・With

thesameriskaverseness，thelargertheunitcostc is，thelesstheimpactontheobjectivefunctionis．

5 Conclusion

Inthispaper，WepreSentedastochasticmodel

foroptimizingbandwidtha1locationinCommuni−

Cation Networks（CNs）and derived the optimal

bandwidth capacity with thc penalty cost・We

alsoanaly2：edtheriskaversenessinCNsunderthe

mean−Variance framework．N11mericalresult，S・re−

Vealedtheimpactsofthepenaltycostandtherisk

aversenessonthenetworkperformance・

Acknowledgments

ThisworkwassupportedinpartbyGRANT−

IN−AID FORSCI．RES．（No．16560350） d l ▼▲ a VJ V﹀ b b

MEXT．ORC（2004−2008），Japan andin part

NSFCandMADIS，China． Refbrences 【1］D・MitraandQ．Wang，“StochastictrafRcengineer− 1ng，With applicationstonetworkrevenueman− agement，”Proc．L？fIEEE〃膵OCOM，2003・【2］D・MitraandQ・Wang，“Risk−aWarCnCtWOrkprofit managementinatwo−tiermarket，”Proc．qf18th